"The Thracian(s), Orpheus, and Linus shall not surpass me in songs, although the mother to this Orpheus, Calliopea, and the father to this Linus, handsome Apollo, may help/support [them]."

A few assumptions I have made are:

1. the nec...nec cancel out and become positive because of the non, which I believe is possible from Gildersleeve 449. 2. the huic Orphei...huic Lino are in the dative because adesse can take the dative. I also think that the huics may not need to be included in the translation because they only function as a placeholder for the substantive they represent, and this allows the metre to work -- similar to Catullus 100.

"Neither Thracian Orpheus nor Linus will defeat me in [a contest of] song[s], even if [his] mother is present to support [adsit + dative] this one or his father is present to support this one, Calliope [for] Orpheus, handsome Apollo [for] Linus."

Here the double negatives don't cancel each other out. Otherwise, Vergil would be saying that he would lose the contest. See Allen & Greenough sec. 327: